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Gromov-Witten invariants on Grassmannians. (English) Zbl 1063.53090
The main result of the paper under review is: any three-point genus zero Gromov-Witten invariant on a Grassmannian \(X\) is equal to a classical intersection number on a homogeneous space \(Y\) of the same Lie type. The statement was proven when \(X\) is a type \(A\) Grassmannian, and, in types \(B\), \(C\), and \(D\), when \(X\) is the Lagrangian or orthogonal Grassmannian parametrizing maximal isotropic subspaces in a complex vector space equipped with a non-degenerate skew-symmetric or symmetric form. Their key identity for Gromov-Witten invariants is based on an explicit bijection between the set of rational maps counted by a Gromov-Witten invariant and the set of points in the intersection of three Schubert varieties in the homogeneneous space \(Y\). It should be noted that the proof of their result does not use moduli spaces of maps and only uses basic algebraic geometry. In types \(B\), \(C\), and \(D\), their result may be used to give new proofs of the structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians obtained by A. Kresch and H. Tamvakis in [J. Algebr. Geom. 12, No. 4, 777–810 (2003; Zbl 1051.53070) and Compos. Math. 140, No. 2, 482–500 (2004; Zbl 1077.14083)]. Their methods can also be used to prove a quantum Pieri rule for the quantum cohomology of sub-maximal isotropic Grassmannians.

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14N15 Classical problems, Schubert calculus
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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